A particular class of fractures in the Earth develops as a result of internal pressurization by a viscous fluid. These fractures are either man-made hydraulic fractures created by injecting a viscous fluid from a borehole, or natural fractures such as kilometers-long volcanic dikes driven by magma coming from the upper mantle beneath the Earth's crust. Man-made hydraulic fracturing “treatments” have been performed for many decades, and for many purposes, including the recovery of oil and gas from underground hydrocarbon reservoirs.
Despite the decades-long practice of hydraulic fracturing, many questions remain with respect to the dynamics of the process. Questions such as: how is the fracture evolving in shape and size; how is the fracturing pressure varying with time; what is the process dependence on the properties of the rock, on the in situ stresses, on the properties of both the fracturing fluid and the pore fluid, and on the boundary conditions? Some of the difficulties of answering these questions originate from the non-linear nature of the equation governing the flow of fluid in the fracture, the non-local character of the elastic response of the fracture, and the time-dependence of the equation governing the exchange of fluid between the fracture and the rock. Non-locality, non-linearity, and history-dependence conspire to yield a complex solution structure that involves coupled processes at multiple small scales near the tip of the fracture.
Early modeling efforts focused on analytical solutions for fluid-driven fractures of simple geometry, either straight in-plane strain or penny-shaped. They were mainly motivated by the problem of designing hydraulic fracturing treatments. These solutions were typically constructed, however, with strong ad hoc assumptions not clearly supported by relevant physical arguments. In recent years, the limitations of these solutions have shifted the focus of research in the petroleum industry towards the development of numerical algorithms to model the three-dimensional propagation of hydraulic fractures in layered strata characterized by different mechanical properties and/or in-situ stresses. Devising a method that can robustly and accurately solve the set of coupled non-linear history-dependent integro-differential equations governing this problem will advance the ability to predict and interactively control the dynamic behavior of hydraulic fracture propagation.
For the reasons stated above, and for other reasons stated below which will become apparent to those skilled in the art upon reading and understanding the present specification, there is a need in the art for alternate methods for modeling various behaviors of hydraulic fracturing operations.